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Simplifying: Putting it All Together

To simplify an expression means to tidy it such that it has as few parentheses and as few terms as possible. Ideally, it would be nice if the entire thing vanished into oblivion. That would make our lives much easier.

The way we simplify is by eliminating parentheses and combining like terms until there are no parentheses left and no like terms left to combine.

Sample Problem

Simplify 4{3 + (x - 1)}.

First way: Work from the inside out. We don't actually need those innermost parentheses. Get rid of them. They are like wisdom teeth or a Blockbuster Video gift card—completely useless and unnecessary.

4{3 + x - 1}.

Now combine like terms to get

4{2 + x}.

Finally, multiply using the distributive property to get

8 + 4x.

Second way: Work from the outside in. First, use the distributive property to get

12 + 4(x - 1).

Then, use the distributive property again to get

12 + 4x - 4,

and finally, combine like terms to find that 8 + 4x.

Okay, we're all combined out now. Break time.

Be Careful: Simplifying stuff in parentheses before you use the distributive property will often save you work. It is also a good exercise in applying the Order of Operations we learned before. Remember Please Excuse My Aunt, or Please Excuse My Dear Aunt Sally for those who are really pedantic. Notice that, in the second way of answering the example above, the distributive property needed to be used twice instead of once. But, you know what they say: "Using the distributive property twice is just as nice!" we are not exactly sure whom the "they" is referring to, but somebody somewhere must say it. 

Sample Problem

Show that the area of the trapezoid pictured below is given by 

First, let's chop the trapezoid into pieces (aw, don't feel bad—it has no nerve endings), and re-label some lengths. The lower base of the trapezoid has length b2, which we can rewrite as x + b1 + (b2 - b1 - x).

The area of the trapezoid is the area of the triangle on the left side, plus the area of the rectangle, plus the area of the triangle on the right side. Unfortunately, there is no magical formula for finding the area of a trapezoid at one fell swoop—we need to do it piecemeal. Using the area formulas for triangles and rectangles, we get

Ay, chihuahua! We sure would love to simplify that bad boy. Applying the distributive property gets us

.

Now combine like terms, remembering that b1h = hb1 to find that

,

and finally factor out to get

Couldn't be more straightforward. Well, it could, but then it wouldn't be as much fun.

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